By Manjul Bhargava (auth.), Claus Fieker, David R. Kohel (eds.)
From the reviews:
"The ebook includes 39 articles approximately computational algebraic quantity conception, mathematics geometry and cryptography. … The articles during this ebook replicate the huge curiosity of the organizing committee and the individuals. The emphasis lies at the mathematical conception in addition to on computational effects. we propose the ebook to scholars and researchers who are looking to examine present examine in quantity conception and mathematics geometry and its applications." (R. Carls, Nieuw Archief voor Wiskunde, Vol. 6 (3), 2005)
Read Online or Download Algorithmic Number Theory: 5th International Symposium, ANTS-V Sydney, Australia, July 7–12, 2002 Proceedings PDF
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Additional resources for Algorithmic Number Theory: 5th International Symposium, ANTS-V Sydney, Australia, July 7–12, 2002 Proceedings
49–105. Ser97. , Friedr. Vieweg & Sohn, Braunschweig, 1997, Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt, With a foreword by Brown and Serre. Shl89. Alexandra Shlapentokh, Extension of Hilbert’s tenth problem to some algebraic number ﬁelds, Comm. Pure Appl. Math. 42 (1989), no. 7, 939–962. Shl92. Alexandra Shlapentokh, Hilbert’s tenth problem for rings of algebraic functions in one variable over ﬁelds of constants of positive characteristic, Trans. Amer.
For the sake of simplicity, assume that q is prime. In that case, λq−3 ≡ λ−2 (mod q). Moreover, thanks to the relation i j i+j+1 tˆ(φ(g λ ), φ(g λ )) = g λ , q−3 −2 it is easy by using addition chains to compute Λ = g λ = g λ . Remarking λab ab ˆ that t(φ(g ), φ(Λ)) = g we can now solve the CDH problem in G3 = G1 (and also in G2 ) with two applications of the pairing tˆ. As a consequence of this digression, we can now remark that the hardness of the Tate Diﬃe–Hellman problem implies that the Tate pairing is hard to invert when one side of the pairing is ﬁxed.
193 (2000), no. 2, 463–500. Shl00b. Alexandra Shlapentokh, Hilbert’s tenth problem over number ﬁelds, a survey, Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Ghent, 1999), Amer. Math. , Providence, RI, 2000, pp. 107–137. Sil92. Joseph H. Silverman, The arithmetic of elliptic curves, Springer-Verlag, New York, 1992, Corrected reprint of the 1986 original. Vid94. Carlos R. Videla, Hilbert’s tenth problem for rational function ﬁelds in characteristic 2, Proc. Amer. Math.